where \(\operatorname{Out}(\mathbb {F}_g)\)  denotes the outer automorphism group of the free group. By a theorem of Laudenbach [1]}, this homomorphism is surjective and its kernel is a finite abelian group generated by twists along embedded spheres in \(W_g\) . Now, \(\operatorname{Out}(\mathbb {F}_g)\)  is of type \(F_\infty \)  by work of Culler–Vogtmann [2]}, and hence Proposition REF  gives:
